Tool-Narayanaswamy-Moynihan Research Articles

Enthalpy relaxation of polymers: comparing the predictive power of two configurational entropy models extending the AGV approach

The Tool-Narayanaswamy-Moynihan (TNM) phenomenological model is widely accepted in order to describe the structural relaxation of glasses. However several quantitative discrepancies can be found in the literature that cannot be entirely ascribed to the experimental errors. In this work we compare the predictive power of two recently proposed configurational entropy approaches extending the TNM formalism. Both of them change the treatment of non linearity by adding a free parameter. We use Differential Scanning Calorimetry (DSC) experiments in order to test the models in two different polymers. One of them is a commercial PMMA sample, the other is a side chain liquid crystal azo-benzene polymer properly synthesized for optical nanorecording purposes. Different results were found for the two systems. In the PMMA sample only one of the new models was able to improve the agreement between DSC experiments and theory with respect to the TNM model, whereas in the second polymer both the approaches were able to describe the experiments better than TNM model.

Enthalpy recovery in low molecular weight PMMA

The phenomenological model of Tool–Narayanaswamy–Moynihan (TNM), has been widely used in order to describe enthalpy and volume relaxation experiments in different glass forming systems. However some difficulties were evidenced, especially in polymeric samples. In this work we study the enthalpy relaxation mechanism of a low molecular weight poly(methyl methacrylate) (PMMA) sample via differential scanning calorimetry experiments. We show that the TNM model, with the Scherer–Hodge expression for relaxation times, describes the aging mechanism fairly well so that it was possible to simultaneously reproduce different experimental scans with a single set of parameters. Furthermore the long term enthalpy lost after annealing was found to agree with the prediction of the model. On the contrary, from different papers in the literature it is found that the TNM model shows severe difficulties in describing the enthalpy recovery mechanism of high molecular weight PMMA samples. These results could support the idea that chain entanglements play a role in the difficulties encountered by TNM model in high polymers as suggested by others in the past.

Enthalpy Relaxation of Layered Silicate‐Epoxy Nanocomposites

AbstractUsing standard and temperature‐modulated differential scanning calorimetry, we have studied the segmental relaxation behavior of montmorillonite‐epoxy nanocomposites. The experimental results showed that the incorporation of clay particles caused the epoxy network to exhibit slower relaxation dynamics compared with the neat epoxy resin. Analysis based on the Tool‐Narayanaswamy‐Moynihan (TNM) model indicated that the nanocomposite systems had greater nonlinearity, higher apparent activation energy, and broader relaxation time distribution than the neat epoxy resin. Further analysis based on the Adam‐Gibbs model indicated that the presence of clay nanoparticles decreased the macroscopic configurational entropy of the epoxy network, and increased the size of the cooperative rearrangement region as well as the fundamental energy barrier. The peculiar relaxation behavior observed in the present experiments was attributed to the confinement effect of nanoparticles on molecular entities during the segmental relaxation. This confinement effect led to a stronger temperature dependence of the relaxation behavior near the glass transition temperature, or greater fragility. An inverse correlation between nonexponentiality and fragility was also observed, which is consistent with the results reported in the literature.The original MDSC curves: complex heat capacity (C_{\rm P}^*) versus temperature; the inset shows the nonexponentiality parameters (β) for all the samples.magnified imageThe original MDSC curves: complex heat capacity (C_{\rm P}^*) versus temperature; the inset shows the nonexponentiality parameters (β) for all the samples.

Glass transition of thin films of poly(2-vinyl pyridine) and poly(methyl methacrylate): nanocalorimetry measurements

Glass transitions were observed in thin films of poly(2-vinyl pyridine) (P2VP) and poly(methyl methacrylate) (PMMA) using a scanning nanocalorimetry technique which has both high sensitivity (10 −9 J/K) and high scan rates (10 4–10 5 K/s). Samples were deposited by the spin-cast method. The thickness of samples was 100–400 nm. Glass transition temperature, obtained by nanocalorimetry, is shifted toward higher temperatures by 10–20 K and activation enthalpy of glass transition is shifted to lower values by factor of 2–4. The glass transition characteristics of both polymers are discussed in terms of the standard Tool–Narayanaswamy–Moynihan (TNM) multi-parameter model.

Studying the Glass Transition by DSC and TMDSC

First, the principal features of the glass transformation process in polymers are reviewed, and then it is shown how they are manifest in conventional DSC, and the quantitative analysis of typical DSC data is discussed in terms of the Tool-Narayanaswamy-Moynihan (TNM) model. Subsequently, the way in which the glass transition is manifest in Temperature Modulated DSC is presented, and the effects of both experimental and material parameters are discussed. In conclusion, the two techniques are compared in terms of the information they provide about the glass transformation process.

The effect of non-linearity contribution to the volume and enthalpy relaxations in amorphous materials

The effect of non-linearity contribution to the relaxation behavior of amorphous materials is analyzed within the Tool–Narayanaswamy–Moynihan (TNM) model. It is shown that the rate of relaxation response to a temperature jump can conveniently be characterized on the basis of the stabilization period and the fictive relaxation rate. A simple equation relating these quantities and parameters of the TNM model is derived. This equation separates the contribution of non-exponentiality and non-linearity. It has been tested for polyvinylacetate and As 2S 3 relaxation data. Within the limits of experimental error it seems that volume and enthalpy relaxations in these materials exhibit a similar response that can be well correlated with previously published TNM parameters.

On the application of the Adam–Gibbs equation to the non-equilibrium glassy state

A new extension of the Adam–Gibbs (AG) equation to the non-equilibrium glassy state is presented, whereby the configurational entropy is a function of both fictive temperature and thermodynamic temperature. The relative contributions of each are determined by an entropic non-linearity parameter x s analogous to the non-linearity parameter x of the Tool–Narayanaswamy–Moynihan (TNM) equation. This new form of the AG equation resolves many inconsistencies in the TNM approach: for example, the dependence of x on annealing temperature and time observed by many workers, as well as often unreasonable values for x and for T 2. Furthermore, it can rationalise the frequently reported correlation between x and β, the KWW non-exponentiality parameter. A physical interpretation of x s is made in terms of the Gibbs–DiMarzio theory and the relative importance of the freezing-in of either holes or of high energy bond conformations.

The effect of non-linearity contribution on the volume and enthalpy relaxation in amorphous materials

The effect of non-linearity contribution to the relaxation behavior of amorphous materials is analyzed with the Tool–Narayanaswamy–Moynihan (TNM) model. It is shown that the rate of relaxation response to a temperature jump can conveniently be characterized on the basis of the stabilization period and the fictive relaxation rate. A simple equation relating these quantities and parameters of the TNM model is derived. This equation separates the contribution of non-exponentiality and non-linearity. It has been tested for PVA and As 2S 3 relaxation data. Within the limits of experimental error it seems that volume and enthalpy relaxation in these materials exhibit a similar response that can be well correlated with previously published TNM parameters.

The application of temperature-modulated DSC to the glass transition region: II. Effect of a distribution of relaxation times

An analysis of temperature-modulated differential scanning calorimetry (TMDSC) in the glass transition region is presented. It extends an earlier and simpler model by introducing a distribution of relaxation times, characterised by a Kohlrausch–Williams–Watts (KWW) stretched exponential parameter β, in addition to the usual kinetic parameters of relaxation, namely the Tool–Narayanaswamy–Moynihan (TNM) non-linearity parameter x and the apparent activation energy Δ h ∗. The present model describes, more realistically than did its predecessor, all the characteristic features of TMDSC in the glass transition region, and it has been used to examine the effects of the important experimental variables, namely the period of modulation and the underlying cooling rate. It is shown that, for typical experimental conditions in practice, it is likely that there well be an interaction between the vitrification process, due to the underlying cooling rate, and the dynamic glass transition whereby the complex heat capacity C p ∗ shows a sigmoidal decrease in a temperature range dependent on the modulation frequency. Accordingly, care must be exercised in the quantitative evaluation of TMDSC data in the glass transition region, and suggestions are made regarding the optimum procedures in this respect. Also, by comparing the cooling rate and modulation period required to define the same transition temperature for conventional DSC and C p ∗ , respectively, a correspondence between them is obtained which allows the magnitude of temperature fluctuations in Donth’s fluctuation dissipation theorem to be evaluated. Finally, it is shown that β and x have similar effects on conventional DSC cooling curves, but have very different effects on C p ∗ , whereby there is little effect of x but a significant broadening of the transition as β decreases. It is argued that the breadth of the C p ∗ transition therefore provides a measurement of β independent of the value of x, thus resolving a problem that has existed for some years.

Application of an extended Tool–Narayanaswamy–Moynihan model: Part 1. Description of vitrification and complex heat capacity measured by temperature-modulated DSC

Complex heat capacity in equilibrium can be considered as a compliance in the scheme of linear response. Nevertheless, often the Tool–Narayanaswamy–Moynihan (TNM) or the Kovacs–Aklonis–Hutchinson–Ramos (KAHR) models are used to describe complex as well as total heat capacity in the glass transition region. This is attractive because the TNM–KAHR models take into account the dependence of the relaxation time on temperature as well as fictive temperature and are able to handle non-equilibrium situations. A TNM-model extended by a non-Arrhenius temperature dependence of the relaxation time is applied to describe results from temperature-modulated DSC (TMDSC) measurements. The temperature dependence of the relaxation time, as introduced in the model, is independently determined by a VFTH-fit to heat capacity spectroscopic data in a wide frequency range. Finally, only the stretched exponential factor β and the non-linearity parameter x of the TNM-model must be adjusted to describe the measured curves reasonable well. The model is capable to describe all features of the heat capacities measured in TMDSC scan experiments in the glass transition region of polystyrene. This includes the curve shape, the frequency dependence of T g( ω), the cooling rate dependence of T g( q 0) and possible interference between vitrification and dynamic glass transition. For the complex heat capacity in equilibrium, above vitrification, no influence of fictive temperature on the relaxation time was observed. Consequently, it can be considered as an equilibrium quantity.

Structural relaxation near the glass transition temperature

Enthalpy relaxation processes on a model bulk glass, As 2Se 3, were studied below the glass transition temperature, T g, as well as at temperatures close to the glass transition overshoot. Extents of relaxation processes below T g were measured by enthalpy of endothermic relaxation peak. Relaxation processes above T g were determined by melting enthalpies of partly crystallized glass during relaxation. We found an exponential temperature dependence of physical aging. Completely annealed glasses reach the equilibrium having excess of enthalpy compared to the equilibrium supercooled liquid at the same temperature. Using the Tool–Narayanaswamy–Moynihan (TNM) model, we found that the fictive temperature, T f, decreased during isothermal annealling toward aging temperature, T a, and at equilibrium is equal to annealing temperature, T a. When the glass comes to equilibrium the distribution of relaxation times decreases. The description of relaxation processes at the temperatures slightly above T g is more complicated. The nuclei are created in the first step. In the second one these nuclei can create thermodynamically non-equilibrium agglomerates, which consequently can decrease their energy by becoming amorphous.

Application of the Adam−Gibbs Equation to the Non-Equilibrium Glassy State

The Tool-Narayanaswamy-Moynihan (TNM) equation for the temperature (T) and fictive temperature (T f ) dependence of the relaxation time in glassy materials is compared with the usual nonlinear form of the Adam-Gibbs (AG) equation. It is shown that the relationship derived between the Narayanaswamy parameter x and the temperature T 2 at which the configurational entropy reduces to zero, namely x 1 - T 2 /T f , leads to unrealistic values of T 2 for many polymer glasses. This problem is resolved by expressing the configurational entropy as a function of both T and T f , with a partitioning parameter x s (0 ≤ x s ≤ 1) controlling their respective contributions. Comparing TNM with this new nonlinear AG expression incorporating S c (T,T f ) leads to an explicit relationship between x and x s involving T, T 2 , and T f , from which a number of predictions may be made. (1) For T T f , i.e., for relaxations close to equilibrium, the quantity 1 - T 2 /T f is identified as the minimum possible value for x, implying that T 2 ≥ T f (1 - x), by an amount depending on the value of x s . This resolves the apparently anomalous values of T 2 . (2) For relaxations further from equilibrium, the TNM equation with constant x becomes increasingly inappropriate. (3) With increasing annealing temperature and increasing annealing time, the analysis predicts increasing values of x, as has often been reported in the literature. The origin of the dependence of S c on T and T f is considered from the theory of Gibbs and DiMarzio, and it is argued that typical values ofx observed experimentally may be associated with the freezing-in of only a certain fraction of either flexed bonds and/or vacant lattice sites (holes) at the glass transition. Thus, it is possible to identify X s , and indirectly x, with the relative contributions of physically meaningful parameters, such as intermolecular and intramolecular bond energies, to the freezing-in process.

Quantitative analysis of errors in TMDSC in the glass transition region

Temperature modulated differential scanning calorimetry (TMDSC) data are simulated through the glass transition region using the Tool–Narayanaswamy–Moynihan (TNM) model of structural recovery. The simulated data are analyzed using a fast Fourier transform similar to conventional Temperature-modulated differential scanning calorimetry techniques. The glass transition temperature ( T g) and enthalpy of aging (Δ H a) were calculated from the simulated data and compared with the values expected based on linear isothermal simulations for the case of the glass temperature and based on the input simulated data for Δ H a. The errors associated with the TMDSC data analysis were thus quantified.

Enthalpy recovery in polymeric glasses: Does a correlation exist between the empirical (T‐N‐M) relationship parameters and the glass structures?

AbstractTo characterize the enthalpy recovery which appears during physical aging in glassy materials and especially in glassy polymers, the empirical Tool‐Narayanaswamy‐Moynihan (TNM) relationship for the relaxation time is widely used. This relationship depends essentially on three parameters, the apparent activation energy Δh*, the non linearity parameter x and the fictive temperature Tf. Even if a great amount of data (x, Δ*) can be found in the literature, no direct correlation between the values of these parameters and the engaged structures are available. On the one hand, using some polymers like Poly(ethylene terephthalate), Polymethyl(α‐n‐alkyl)acrylates and three dimensional epoxy networks for which structural changes are controlled and on the other hand, using some recently published results concerning other materials including inorganic polymers (Ge‐Se and Ge‐Te glasses), we show that it is possible to propose some interesting tendencies. In particular it is shown that the value of x depends on the intermolecular strength for organic polymers and on the connectivity for inorganic polymer.

Physical aging in poly(ethylene oxide)/ atactic-poly(methyl methacrylate) blends

Physical aging in poly(ethylene oxide)(PEO)/ atactic-poly(methyl methacrylate) ( a-PMMA) blends with a low content of PEO is discussed. Behaviour in the T g-region is analysed. Different compositions are prepared by mixing in the melt, and were compared at the same temperature distance to mid-point enthalpic T g. Macrolevel properties (enthalpy and external volume) are discussed in terms of the Tool–Narayanaswamy–Moynihan (TNM) model. In particular, the non-linearity x parameter (determined using the peak shift method), the apparent activation enthalpy Δ h * , and the combination θ·(1– x), where θ=Δ h */ R· T g 2, are used to discuss overall segmental co-operativity and relaxation in the blends. Based on aging in viscoelastic properties (aging rate), structural mobility is discussed. From obtained results, an increase of co-operativity, and decrease of both, relaxation rate and change of structural mobility, with the increasing content of PEO in the blend, is concluded for temperatures between the dilatometric and enthalpic T g. Agreement with data from studies at the microlevel by other authors, where an onsetting with time phase separation in PEO/PMMA has been postulated, is found.

A phenomenological study of the structural relaxation of an inorganic glass (Li 2O2SiO 2)

The structural relaxation process of an inorganic glass (Li 2O2SiO 2) has been studied by differential scanning calorimetry. The sample is subjected to different thermal ageing histories with isothermal stages at an ageing temperature of T g −30°C for different ageing times and at an ageing time of 16 h for different ageing temperatures. A four-parameter Tool-Narayanaswamy-Moynihan (TNM) model is applied to simulate the normalized specific-heat curves measured. The ageing-temperature and ageing-time dependence of the structural relaxation parameters in the TNM model is obtained.

Structural relaxation of As 2S 3 glass by length dilatometry

Structural relaxation of As 2S 3 glass was studied by dilatometry. The Tool–Narayanaswamy–Moynihan (TNM) model was successfully applied to the quantitative description of isothermal experiments performed within 40°C below T g. The activation energy of the relaxation process is identical within the experimental error with that of viscous flow, which was found to be 267 kJ/mol. The non-linearity and non-exponentiality parameters were evaluated as x=0.31 and β=0.82, respectively. The temperature dependence of the normalized volume relaxation rate agrees well with the prediction based on the TNM model. This approach can be used for comparison of structural relaxation kinetics in various non-crystalline materials.

The effects of structural recovery and thermal lag in temperature-modulated DSC measurements

It is well known that structural recovery in polymeric glass formers leads to the enthalpy overshoot in nonisothermal experiments. One common model for describing this phenomenon is the Tool-Narayanaswamy-Moynihan (TNM) equation. Here we apply the TNM equation to analyze a typical temperature-modulated DSC (TMDSC) trace and examine the influence of the material nonlinearities on the dynamic heat flow. We show that, in the glass-transition region, the oscillating heat-flow response to the sinusoidal temperature input is a distorted sine wave. Hence, linear analysis of the phase lag between heat flow and temperature is not physically meaningful. The degree of distortion of the sinusoidal heat flow increases as the magnitude of the excess enthalpy annealing peak increases. In addition, we perform a thermal analysis of samples having different geometries and show that an apparent phase lag can result in the measured response due to the presence of thermal gradients in the sample. The significance of the results is discussed.

The effect of structural parameters on the enthalpy relaxation of an inorganic glass (Li 2O · 2SiO 2)

The structural relaxation process of an inorganic glass (Li 2O · 2SiO 2) at an ageing temperature of 703 K for an ageing timeof 1 h has been studied by differential scanning calorimetry. A four-parameter model—the Tool-Narayanaswamy-Moynihan (TNM)—model was applied to simulate the normalized specific heat curve measured. A set of optimized parameters, Δ h */ R, β, In A, and x was obtained. Then the effects of variation of each adjustable parameter on the calculated specific heat were summarized.